A train leaves Pune at 7 :30 am and reaches Mumbai at 11 : 30 am. Another train leaves Mumbai at 9 : 30 am and reaches Pune at 1 : 00 pm. Assuming that the two trains at constant speeds, at what time do the two trains cross each other-

Options

A. `10 : 20am`
B. `11:30am`
C. `10:26am`
D. Data not sufficinet

Explanation

To solve the problem of when the two trains cross each other, we will follow these steps:

### Step 1: Determine the speeds of both trains.

1. **Train 1 (Pune to Mumbai)**:
- Departure: 7:30 AM
- Arrival: 11:30 AM
- Total time taken = 4 hours
- Speed of Train 1 = Distance (D) / Time = D / 4 km/h

2. **Train 2 (Mumbai to Pune)**:
- Departure: 9:30 AM
- Arrival: 1:00 PM
- Total time taken = 3.5 hours
- Speed of Train 2 = Distance (D) / Time = D / 3.5 km/h

### Step 2: Calculate the distance covered by Train 1 by the time Train 2 starts.

- Train 1 travels from 7:30 AM to 9:30 AM (2 hours).
- Distance covered by Train 1 in 2 hours = Speed × Time = (D / 4) × 2 = D / 2 km.
- Remaining distance for Train 1 to reach Mumbai = D - D/2 = D/2 km.

### Step 3: Set up the equation for when the two trains meet.

- After 9:30 AM, both trains are moving towards each other.
- Let the time after 9:30 AM when they meet be T hours.
- Distance covered by Train 1 in T hours = Speed × Time = (D / 4) × T.
- Distance covered by Train 2 in T hours = Speed × Time = (D / 3.5) × T.

### Step 4: Write the equation for the total distance.

Since they meet after covering the remaining distance:
\[ \text{Distance covered by Train 1} + \text{Distance covered by Train 2} = \text{Remaining distance} \]
\[ \left(\frac{D}{4} \cdot T\right) + \left(\frac{D}{3.5} \cdot T\right) = \frac{D}{2} \]

### Step 5: Simplify the equation.

1. Factor out D (assuming D ≠ 0):
\[ \frac{T}{4} + \frac{T}{3.5} = \frac{1}{2} \]

2. To solve for T, find a common denominator for the left side:
- The common denominator of 4 and 3.5 is 14.
- Rewrite the equation:
\[ \frac{3.5T + 4T}{14} = \frac{1}{2} \]

3. Combine the terms:
\[ \frac{7.5T}{14} = \frac{1}{2} \]

4. Cross-multiply to solve for T:
\[ 7.5T = 7 \]
\[ T = \frac{7}{7.5} = \frac{14}{15} \text{ hours} \]

### Step 6: Convert T to minutes.

1. Convert \(\frac{14}{15}\) hours to minutes:
\[ \frac{14}{15} \times 60 = 56 \text{ minutes} \]

### Step 7: Calculate the meeting time.

1. The trains meet 56 minutes after 9:30 AM:
- 9:30 AM + 56 minutes = 10:26 AM.

### Final Answer:

The two trains cross each other at **10:26 AM**.

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